Acta Universitatis Palackianae Olomucensis. Facultas Rerum
Naturalium. Mathematica
Ivan Chajda; Miroslav Haviar
Induced pseudoorders
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 30 (1991), No.
1, 9--16
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ACTA UNIVERSITATIS PALACKIANAE OLOMUCENSIS
FACULTAS RERUM NATURALIUM
1991
MATHEMATICA
XXX
VOL. 100
Katedra algebry a geometrie
přírodovědecké fakulty Univerzity Palackého v Olomouci
Vedoucí katedry: Doc . RNDr . Jiří Rachůnek, CSc.
INDUCED PSEUDOORDERS
IVAN CHAJDA, MIROSLAV HAVIAR
(Received January 10, 1990)
Abstract: Let R be a reflexive binary relation on a set A.
We proceed to show under which conditions the relation
= ROR~
is an equivalence on A and the factor relation
R
T( ) R/T(R)
is a pseudoorder on the factor set A / T ( R ) .
Key words: quasiorder, order, pseudoorder, tolerance, re­
lation, equivalence relation.
MS Classification
: 04A05 , 06A99
Let A be a non-void set. Let R be a binary relation on A
and
£ be an equivalence on A such that £ SL R. Denote by R/£
the binary relation defined on the factor set A/£
B,C£A/£
, <B,C^€R/£
b€B
, c€C
by the rule:
if and only if there exist elements
such that
<b,c>€ R .
By a quasiorder on a set A is meant a reflexive and transitive binary relation on A. An order on A is a reflexive,
antisymmetrical and transitive relation on A. The following
elementary proposition is well-known:
- 9 -
Proposi tion. Let Q be a quasiorder on a set A / p
£ (Q) = Q A Q "
is an equivalence on A (evidently
. The relation
£(Q)£- Q) and
the relation Q/£ (Q) is an order on the factor set hit
(Q).
Since an order and quasiorder on A are transitive binary
relations, we will try what happens if the transitivity of Q in
the Proposition would be omitted.
A binary relation P on a set A / 0 is called a pseudoorder
if P is reflexive and antisymmetrical. A binary relation T on A
is called a tolerance if it is reflexive and symmetrical.
Clearly, every order is a pseudoorder (but not vice versa) and
every equivalence is a tolerance (but not vice versa, see e.g.
[2], [4]), Denote by 00 the identical relation on A, i.e.
< a , b > € o o if and only if a = b.
Definition 1. Let T be a tolerance on a set A / 0. A non-void
subset B - A is called a block of T if B is a maximal subset
of A such that x,y€B
implies <x, y > € T. Denote by A/T the set
of all blocks of T.
For the concept of block and properties of A/T, see e.g.
[l] and [3J. It is evident that if T is an equivalence on A,
the concept of equivalence class coincides with the concept of
block and A/T is the factor set.
For a binary relation R on A, denote by T*(R) = R O R " .
The following lemma is evident:
Lemma 1. Let R be a reflexive relation on a set A. Then
(i)
?^(R) is a tolerance on A;
(ii)
if T is a tolerance on A, then
i (T) = T.
Definition 2. Let R be a binary relation on a set A and T be
a tolerance on A such that T -=• R. The relation R/T defined on
the set A/T by the rule:
(a) B,C€A/T
, < B , C > 6 R / T if and only if there exist elements
b£ B , c€ C
with
<b,c>£ R
will be called induced by R on A/T.
Hencefore, we will try under which conditions, the concepts
of a quasiorder, an equivalence and an order in the Proposition
can be replaced by concepts of a reflexive relation, a tolerance
and a pseudoorder, respectively.
- 10 -
Definition 3. Let R be a binary relation on A. R is called
weakly transitive if for each three elements a,b,c of A,
< a , b > € T(R) and <b,c> 6 T ( R )
imply
<a , c > e T ( R ) •
Lemma 2. For a reflexive relation R on a set A, the foll°w--n9
conditions are equivalent:
(a)
T(R) is an equivalence on A;
(b)
R is weakly transitive.
The proof follows immediately from Definition 3 and Lemma 1.
Example 1. Let A be a three element set {a,b,cl and R be a
reflexive relation on A given by
R = cOU{<a5b>, <b,c>,<c,a>{
,
(
X
-*• •
denoíes
<x<y>eR and lh<z
reffzyt'vz
paCrQ are
omtiiecL )
>•.
b
•
Fig. 1
a
see Fig.l. Then R is weakly transitive relation which is not
. . .
.
transitive and
<*>-/• r, \
T"(R)
= oO
зt A. If
Lemma 3. Let R be a reflexive binary relation _oп a set
order on A/T(R), then R is weakly transi tive
oп A.
? ( R ) and
the tolerrance
reflexive and b e C as well as b € U , we have <.u,L
< D , C > € R/2"(R).
;rical, thus C = D, i.e. both a.c
However, R/T(R) is antisymmetrical,
ong to the one block of T*(R).
?*(R). Hence <<aa,c
,,cc>>>€€.
T(R)
belong
€ T(R)
the transitivity of the tolerance
proving
Z~(R) and, by Definition 3,
o "the weak transitivity of R.
also
In other words, if we try to give an analogy of the Proition for non-transitive relations, Lemma 3 yields that the
essary condition is that R has to be weakly transitive. The
- 11 -
following example shows that this condition need not be sufficient :
Example 2. Let A = {a,b,cj
and R be a binary relation on A
given by
R
= < o u { < a > b > > <b>a>> < b > c > , < c , a > |
see F i g . 2 .
,
Fig. 2
Then R is reflexive and weakly transitive, i.e.
7T(R) is
an equivalence (its blocks are visualized by dotted lines in
Fig.2). However, A/T(R) is a two element set, see Fig.3, but
Fig. 3
•' {a,b}
R/?(R) is not antisymmetrical, hence R/2"(R) is not a pseudoorder.
Definition 4. A binary relation R on a set A is called semitransitive if for each a,b,c
< a , b > e t(R)
<a,b> c
of A ,
, <b,c>€ R
7*(R) , <c,a> € R
imply <a,c><sR
•
and
imply < c , b > e R .
The situation of Definition 4 can be visualized in Fig.4.
Lemma 4. (i) Every transitive relation is semitransitive;
(ii) Every semitransitive relation is weakly transitive; (iii)
Semitransitive as well as weakly transitive relations satisfy
the Duality Principle, i.e. if R is semitransitive (weakly
transitive) then also R~
has this property.
- 12
The proof is clear and hence omitted.
It is also evident that every antisymmetrical relation is
semitransitive.
If R is a reflexive and weakly transitive relation on A,
then, by Lemma 2, T*(R) is an equivalence on A, thus the block
of T(R) containing an element a € A is uniquely determined. In
such case, denote this block (i.e. the equivalence class of
^(R)) by the symbol a.
Lemma 5. Let R be a reflexive and weakly transitive binary
relation on A. The following conditions are equivalent:
(a) < a , b > € R/r(R)
if
and o n l y i f
<a,b> € R ;
(b) R i s s e m i t r a n s i t i v e .
P r o o f. (a) = > (b): Let <a ,b > e ^(R) and <b,c> S R. By
Lemma 2, £*(R) is an equivalence on A, thus <a,b> € T*(R)
implies ~a = b. Further, <b,c > € R implies < b,c~ > e R/r(R), thui
also < a ,c > €. R/T(R) . By (a), we have < a , c > € R . Analogously
t can be proved for the second law of semitransitivity, thus
it
b) is satisfied.
(b
(b) = > (a): If < a , b > e R then clearly < a,b >6 R/T(R).
It remains to prove the converse implication. Suppose
< a ,b> e R/T(R). Then, by Definition 2, there exist elements
a, e. a and b,6 b such that <a, ,b,> B R. Since R is reflexive,
also a e ¥ , b e b , thus <a,a, > c t*(R), <a,,b,>e R imply by
(b) <a,b 1 >e R. Analogously, < b ] L , b > e ^ ( R ) and <a,b,>€ R
imply <:a,b>€.R proving (a).
Now, we are ready to formulate the answer to the introductory question:
- 13 -
Theorem 1. Let R be a reflexive binary relation on a set A.
The following conditions are equivalent:
(1)
R / « R ) is a pseudoorder on A/?(R) a n d < a , b > € R/^(R)
(2)
R is semitransitive.
if and only if
P r o o f .
<a,b>e R ;
(l) =-> (2): Let a , b , c € A a n d
< b , c > € R . Then < a~,E > <£ R/r(R) and
<a,b>e^(R),
< b ,a > e R/F(R). By (1),
the antisymmetry of R/2"(R) implies a = b. However, < b , c > € R
implies < E ,c > e R/^(R), thus also < a,c > e R/r(R). By (1), we
obtain
<a,c><£ R. Analogously it can be proved the second law
of semitransitivity.
(2) ===-> (1): The second assertion of the condition (1) is
a direct consequence of Lemma 5. It remains to prove the antisymmetry of R/2*(R). By (ii) of Lemma 4 and Lemma 2,
T(R)
is
an equivalence on A. Let < a~,b > <£" R/2~(R) and < E,"a > e R/t(R)
and
.
By Lemma 5, it gives
<a,b>€R
<a,b>€. ^ ( R ) . Since
?*(R) is an equivalence, it implies a = E
< b , a > £ " R , i.e.
proving the antisymetry of R/T(R).
Corollary 1. If R is a reflexive and semitransitive binary
relation on A / 0, then
?*(R) is an equivalence on A and R/^(R)
is a pseudoorder on A/'?(R). Moreover, R/?"(R) is an order on
A/?(R) if and only if R is transitive.
It is clear that if R is not transitive, then R/?"(R) also
is not transitive.
Now, we proceed to show what of the foregoing results can
be transfered from sets into lattices. For this reason, recall
first some other concepts. Let L be a lattice. Denote by — its
lattice order and by V , A its lattice operations join and meet,
respectively. A binary relation R on L is compatible if for any
a,b,c,d of L ,
<a,b>eR
and<c,d>€R
imply < a v c ,
bVd>cR
and
<aAc,
bAd>£R.
It is easy to examine that if R is a reflexive and compatible
binary relation on a lattice L, then 2* (R) is a compatible
tolerance on L (see [lj, [4]) and for any compatible tolerance
T on L, ^ ( T ) = T. By [3], the set L/f(R) forms again a lattice
with the induced lattice order.
- 14 -
By using of Lemma 2, we obtain immediately:
Lemma 6. Let R be a reflexive and compatible binary relation
on a lattice L. The following conditions are equivalent:
(a)
^ ( R ) is a congruence on L;
(b)
R is weakly transitive.
Analogously as in the case of Theorem 1, we can prove the
following
Theorem 2. Let R be a reflexive and compatible binary relation
on a lattice L. The following conditions are equivalent:
(1)
R/?*(R) is a compatible pseudoorder on the lattice
L / M R ) and < a, b > € R/^(T) if and only if
(2)
<a,b>€R;
R is semitransitive.
By [l], a lattice L is tolerance trivial if every compatible tolerance on L is a congruence. The foregoing method of
induced relations enables us "to characterize such lattices:
Theorem 3. A lattice L is tolerance trivial if and only if
T/T = co for every compatible tolerance T on L.
P r o o f .
Let L be tolerance trivial and T be a compatible
tolerance on L. Thus T is a congruence on L, clearly
f ( T ) = T,
thus T/T = T/T(T) = co directly by Definition 2. Conversely,
if T is a compatible tolerance on L and T/f(T) = co
, then
every two distinct blocks of T are disjoint, hence T is a
congruence on L.
We finish our paper by a comparison of the compatible
pseudoorder and the lattice order induced by a reflexive
relation:
Theorem 4. Let R be a reflexive and semitransitive compatible
relation on a lattice L such that
T(R)
— - . The following
conditions are equivalent:
(i) <I,b > e V t t R ) ;
(ii) £ a A b , ~a>e RfT(R)
and < a , a A b > £ R / « R ) ;
(iii) <a A b , a > e < M R ) .
The proof is an easy consequence of the foregoing results
and the fact that a — b in L if and only if a A b = a.
- 15 -
REFERENCES
Ch a j d a, I.: Toleгance tгivial
algebгas and v a г i e t i e s ,
Acta Sci.Math. (Szeged), 46 (1983), 35-40.
[2] Ch a j d a, I.: P a t г i t i o n s , coveгings and Ыocks of
c o m p a t i b l e г e l ә t i o n s , Glasnik Matem. (Zagгeb), 14 (1979),
21-26.
[з] C z é d 1 i, G . : F a c t o г l a t t i c e s by toleгances,
Acta Sci.
Math. (Szeged), 44 (1982), 35-42.
[4] Z e 1 i n k a, B.: Toleгance in algebгaic s t г u c t u г e s ,
Czech.Math.J., 20 (1970), 240-256.
[l]
Ivan Chajda
Department of Algebra and Geometry
Palacký U n i v e r s i t y
Svobody 26, 771 46 Olomouc
Czechoslovakia
Miroslav Haviar
Department of Algebra and Number Theory
Komenský U n i v e r s i t y
Mlýnská dolina, 842 15 Bratislava
Czechoslovakia
Acta UPO, F a c . r e r . n a t . , jJOO, Mathematica XXX (1991) , 9 - 16 -
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